Sector Area Calculator
Compute the area of a pie-slice shaped sector of a circle using its radius and central angle. Ideal for geometry homework, land surveying, and any design work involving circular segments.
About this calculator
A sector is the region of a circle bounded by two radii and the arc between them — like a slice of pie. Its area is a fraction of the full circle's area (πr²), determined by what proportion the central angle is of a full 360°. The formula is: Sector Area = (θ / 360) × π × r², where θ is the central angle in degrees and r is the radius. This can be rewritten using radians as: Sector Area = ½ × r² × θ_rad, where θ_rad = central_angle × π / 180. So the full working formula is: Sector Area = 0.5 × r² × (central_angle × π / 180). This is widely used in engineering, architecture, land measurement, and graphics design when working with circular regions.
How to use
Imagine a pizza slice (sector) cut from a circular pizza of radius 12 inches with a central angle of 45°. Step 1: Convert the angle to radians: 45 × π / 180 = 0.7854 radians. Step 2: Apply the formula: Sector Area = 0.5 × 12² × 0.7854 = 0.5 × 144 × 0.7854 = 56.55 square inches. The pizza slice covers about 56.55 in². Enter radius = 12 and central angle = 45 to verify this in the calculator.
Frequently asked questions
What is the formula for the area of a sector of a circle?
The area of a sector is calculated as Sector Area = ½ × r² × θ, where r is the radius and θ is the central angle in radians. If your angle is in degrees, convert first: θ_rad = degrees × π / 180. Equivalently, you can write it as (central_angle / 360) × π × r². Both forms give the same result and represent the fraction of the full circle's area that the sector occupies.
How is sector area different from segment area?
A sector is the region between two radii and the arc — like a pie slice, including the triangular part near the centre. A segment, on the other hand, is only the region between the chord and the arc, excluding the triangular portion. To find segment area, you subtract the area of the triangle formed by the two radii from the sector area. For most geometry and engineering applications, the sector area is what is needed, but segments arise when calculating areas of circular cutouts or lenses.
Why do I need to convert degrees to radians for sector area?
The standard mathematical formula for sector area uses radians because radians are the natural unit of angular measure in mathematics, directly relating arc length and radius. When angles are given in degrees, multiplying by π/180 converts them to radians before applying the formula. Skipping this step would produce incorrect results by a constant factor. This calculator handles the conversion automatically, so you can simply enter your angle in degrees and receive the correct area.